Certainly! Let's simplify the given expression using the product rule for radicals.
The given expression is:
[tex]\[
\sqrt[6]{\frac{x}{6}} \cdot \sqrt[6]{\frac{7}{y}}
\][/tex]
According to the product rule for radicals, [tex]\(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}\)[/tex]. We can apply this rule to our expression because both radicals have the same index (6th root).
First, identify the expressions under the radicals:
[tex]\[
a = \frac{x}{6}, \quad b = \frac{7}{y}
\][/tex]
Now multiply these two expressions inside a single 6th root:
[tex]\[
\sqrt[6]{\left(\frac{x}{6}\right) \cdot \left(\frac{7}{y}\right)}
\][/tex]
Next, we multiply the expressions inside the radical:
[tex]\[
\frac{x}{6} \cdot \frac{7}{y} = \frac{x \cdot 7}{6 \cdot y} = \frac{7x}{6y}
\][/tex]
Thus, the simplified expression inside the 6th root is:
[tex]\[
\sqrt[6]{\frac{7x}{6y}}
\][/tex]
Therefore, the simplified result of [tex]\( \sqrt[6]{\frac{x}{6}} \cdot \sqrt[6]{\frac{7}{y}} \)[/tex] is:
[tex]\[
\sqrt[6]{\frac{7x}{6y}}
\][/tex]
